Language/Symantic/Lib/Data/Peano.hs

   1 {-# LANGUAGE GADTs #-}
   2 {-# LANGUAGE Rank2Types #-}
   3 {-# LANGUAGE TypeOperators #-}
   4 {-# OPTIONS_GHC -fno-warn-missing-methods #-}
   5 -- | Natural numbers at the type-level, and of kind @*@.
   6 module Language.Symantic.Lib.Data.Peano where
   7
   8 import Data.Type.Equality ((:~:)(Refl))
   9
  10 -- * Types 'Zero' and 'Succ'
  11 -- | Type-level peano numbers of kind '*'.
  12 data Zero
  13 -- | Undefined instance required to build errors.
  14 instance Eq  Zero
  15 -- | Undefined instance required to build errors.
  16 instance Ord Zero
  17 data Succ p
  18 -- | Undefined instance required to build errors.
  19 instance Eq  (Succ p)
  20 -- | Undefined instance required to build errors.
  21 instance Ord (Succ p)
  22
  23 -- * Type 'SPeano'
  24 -- | Singleton for peano numbers.
  25 data SPeano p where
  26  SZero :: SPeano Zero
  27  SSucc :: SPeano p -> SPeano (Succ p)
  28
  29 integral_from_peano :: Integral i => SPeano p -> i
  30 integral_from_peano SZero = 0
  31 integral_from_peano (SSucc x) = 1 + integral_from_peano x
  32
  33 peano_from_integral :: Integral i => i -> (forall p. SPeano p -> ret) -> ret
  34 peano_from_integral 0 k = k SZero
  35 peano_from_integral i k | i > 0 =
  36         peano_from_integral (i - 1) $ \p -> k (SSucc p)
  37 peano_from_integral _ _ = error "peano_from_integral"
  38
  39 peano_eq :: forall x y. SPeano x -> SPeano y -> Maybe (x :~: y)
  40 peano_eq SZero SZero = Just Refl
  41 peano_eq (SSucc x) (SSucc y)
  42  | Just Refl <- x `peano_eq` y
  43  = Just Refl
  44 peano_eq _ _ = Nothing